STOCK PRICES FOLLOW A RANDOM WALK

An empirical evidence

By ANEEL KANWER
Mar 08 - 14, 2004

This article attempts to clarify certain perceptions that stock prices do not follow a random walk. This is accomplished by examining the day of the week effect (Tuesday effect in particular), month effect (Ramazan effect and December effect). The results of the study validate the random walk theory and conclude that there is no effect due to day of week, month as per ANOVA technique used in this article. Other techniques such as Dickey Fuller Test, Box Pierce p-Statistic, Variance Ratio Test, and Unit Root Test are beyond the scope of this article and might provide contradictory results.

Our objective in this article is to validate the random walk hypothesis for the Pakistani Stock Market. We do this by examining the week of day effect and month effect using a simple statistical tool of comparing means.

The study assumes that securities at KSE follow random walk phenomenon. The basis of this study is a discussion held with various market participants who tend to believe that there exists a Tuesday Effect and December Effect.

HYPOTHESIS BUILD FOR ANALYSIS

1. Mean returns are equal on all days.

Ho: Return Monday = Return Tuesday = Return Wednesday = Return Thursday = Return Friday

2. H1: Return January = Return February = Return March = Return April = Return May = Return June = Return July = Return August = Return September = Return October = Return November = Return December

RATIONALE OF THE STUDY

The random walk theory states that the price changes in stocks follow a random pattern. That is, the price changes in stocks are unpredictable; the intense competition amongst the experts ensures that the stock prices already reflect all the relevant information available. Therefore, any price change would occur in response to new information only. This is exactly similar to predicting whether head or tail would occur on each toss of a coin.

The random walk theory states that all information is reflected in the current stock prices, therefore, it can be safely averred that any new information would also take little time to be fully incorporated in the prices, and market participants, thus, would have little time to utilize this new information to realize above normal profits. Because of widespread informational homogeneity every one would be driven to same actions and the above-normal profit potential of new information would quickly be discounted to mere normal level average return offered by the market i.e. the return offered by an average stock; any stock/portfolio that displays significantly varied price change to new information from that of an average stock is offering abnormal return.

To validate the random walk theory a feasible method is to test the variances between the returns offered by individual stocks/portfolios and the market. Any significant variance would mean that there exist inefficiencies and investors can make abnormal profits and the random walk theory does not hold, whereas an insignificant variance between the two values would mean the opposite and would thus validate the random walk theory.

Another way to test the random walk theory would be to see if the current stock prices reflect all available information. This would be accomplished by measuring the speed at which information is incorporated into stock prices and the amount of noise in the price process, which is another domain of research not attempted in this article.

DATA USED FOR THE ANALYSIS

The data consists firstly of the daily closing price data on the scrip traded at Karachi Stock Exchange (KSE) for the period 1998-2003. The data was collected on daily basis from the web site of the KSE. Prices of ten of the top traded stocks and the KSE 100 Index are selected for the study.

The returns are calculated as the logarithmic returns. A 'total returns' index is used in the analysis, that is, daily logarithmic returns dividends reinvested, in order to eliminate any intra-week effect which might be caused by systematic tendencies for shares to go down ex-dividend or pay out dividends on particular days of the week. The price data adjusted for cash and stock dividends and the right issues is used for this analysis.

The daily stock return is defined as the return from the opening (adjusted for stock and cash dividend) of the current trading day to the close of the current trading day.

TIME PERIOD OF STUDY

The period taken for study is June 01, 1998 to May 30, 2003.

ASSUMPTIONS MADE FOR STUDY

The study is made using the following assumptions:

The population is normally distributed.
The variance is same for all the populations
The assumption of homoskedasticity.

STATISTICAL TOOLS FOR ANALYSIS

To test the hypotheses, One Way Analysis of Variance (ANOVA), was used at a significance level of 5%. The acceptance of the hypotheses would show that the mean returns on all the weekdays and months are not significantly different from each other and the rejection would mean that mean returns on at least one day of the week and in at least one month are significantly different from each other.

RESULTS AND CONCLUSIONS

Hypothesis tested: the mean returns on all days of a week are equal.

Ho: Return Monday = Return Tuesday = Return Wednesday = Return Thursday = Return Friday

As can be observed from the Appendix A (except for MPLC) the null hypothesis is accepted, which implies that there is no day-of-the-week effect (for example Tuesday Effect) on the Karachi Stock Exchange.

Hypothesis tested: the mean returns in all months of a year are equal.

Ho: Return January = Return February = Return March = Return April = Return May = Return June = Return July = Return August = Return September = Return October = Return November = Return December.

As can be observed from the Appendix B the null hypothesis is accepted, which shows that there is no month-effect and thus, no Ramazan and/or December effect on the Karachi Stock Exchange.

FURTHER SCOPE OF STUDY

Asset pricing as a whole needs further research especially in context of Pakistani Markets. This study can be further validated using Co-Integration Techniques such as Vector Auto regression or Unit Root Test. Dickey Fuller Test, Box Pierce Q-Statistic and Variance Ratio test are some concepts, still need to be tested on our markets which might provide contradictory results.

APPENDIX A
ANOVA FOR MEASURING EFFECT OF DAY

SYMBOL

SOURCE

SUM OF SQUARES

DF

MEAN SQUARE

F

SIG.

KSE

Between Groups

0.002

4

-

1.519

0.194

 

Within Groups

0.418

1,326

-

-

-

 

Total

0.420

1,330

-

-

-

DGKC

Between Groups

0.021

4

0.005

2.771

0.026

 

Within Groups

2.532

1,338

0.002

-

-

 

Total

2.553

1,342

-

-

-

ENGRO

Between Groups

0.001

4

-

0.356

0.840

 

Within Groups

1.094

1,341

0.001

-

-

 

Total

1.095

1,345

-

-

-

FFCJ

Between Groups

0.007

4

0.002

1.391

0.235

 

Within Groups

1.705

1,329

0.001

-

-

 

Total

1.712

1,333

-

-

-

FFCL

Between Groups

0.002

4

0.001

0.978

0.419

 

Within Groups

0.766

1,341

0.001

-

-

 

Total

0.768

1,345

-

-

-

HUBC

Between Groups

0.003

4

0.001

0.521

0.721

 

Within Groups

1.704

1,340

0.001

-

-

 

Total

1.706

1,344

-

-

-

ICI

Between Groups

0.005

4

0.001

1.340

0.253

 

Within Groups

1.197

1,321

0.001

-

-

 

Total

1.202

1,325

-

-

-

MPLC

Between Groups

0.065

4

0.016

7.816

-

 

Within Groups

2.756

1,322

0.002

-

-

 

Total

2.821

1,326

-

-

-

PSOC

Between Groups

0.005

4

0.001

1.146

0.333

 

Within Groups

1.420

1,341

0.001

-

-

 

Total

1.425

1,345

-

-

-

PTC

Between Groups

0.003

4

0.001

0.920

0.451

 

Whhin Groups

1.130

1,341

0.001

-

-

 

Total

1.133

1,345

-

-

-

SNGP

Between Groups

0.008

4

0.002

1.868

0.114

 

Within Groups

1.515

1,341

0.001

-

-

 

Total

1.523

1,345

-

-

-

APPENDIX B
ANOVA FOR MEASURING EFFECT OF MONTH

KSE100

Between Groups

0.002

11

0

0.58

0.85

 

Within Groups

0.418

1,319

0

-

-

 

Total

0.42

1,330

-

-

-

DGKC

Between Groups

0.015

11

0.001

0.73

0.71

 

Within Groups

2.537

1,331

0.002

-

-

 

Total

2.553

1,342

-

-

-

ENGRO

Between Groups

0.011

11

0.001

1.29

0.23

 

Within Groups

1.083

1,334

0.001

-

-

 

Total

1.095

1,345

-

-

-

FFCJ

Between Groups

0.008

11

0.001

0.55

0.87

 

Within Groups

1.704

1,322

0.001

-

-

 

Total

1.712

1,333

-

-

-

FFCL

Between Groups

0.01

11

0.001

1.58

0.1

 

Within Groups

0.759

1,334

0.001

-

-

 

Total

0.768

1,345

-

-

-

HUBC

Between Groups

0.013

11

0.001

0.94

0.5

 

Within Groups

1.693

1,333

0.001

-

-

 

Total

1.706

1,344

-

-

-

ICI

Between Groups

0.006

11

0.001

0.61

0.82

 

Within Groups

1.196

1,314

0.001

-

-

 

Total

1.202

1,325

-

-

-

MPLC

Between Groups

0.02

11

0.002

0.87

0.57

 

Within Groups

2.801

1,315

0.002

-

-

 

Total

2.821

1,326

-

-

-

PSOC

Between Groups

0.014

11

0.001

1.22

0.27

 

Within Groups

1.411

1,334

0.001

-

-

 

Total

1.425

1,345

-

-

-

PTC

Between Groups

0.003

11

0

0 .28

0.99

 

Within Groups

1.13

1,334

0.001

-

-

 

Total

1.133

1,345

-

-

-

SNGP

Between Groups

0.01

11

0.001

0.8

0.65

 

Within Groups

1.513

1,334

0.001

-

-

 

Total

1.523

1,345

-

-

-